Cardinality constrained index tracking: mixed integer approaches

We consider the problem of index tracking: reproducing the performance of a stock market index, but without purchasing all of the stocks that make up the index. We present two mixed-integer programming formulations of this problem that include a constraint limiting the number of stocks that can be purchased. One approach consists in minimizing the tracking error, which is the variance of the difference between tracking portfolio returns and index returns, the other approach consists in minimizing the CVaR of the negative deviation of the tracking portfolio returns from index returns. We use MATLAB and Gurobi and datasets drawn from real-life stock market indexes.